Symmetric Cauchy stresses do not imply symmetric Biot strains in - PowerPoint PPT Presentation

Symmetric Cauchy stresses do not imply symmetric Biot strains in weak formulations of isotropic hyperelasticity with rotational degrees of freedom. Andreas Fischle ∗ , Patrizio Neff † and Ingo Münch ‡ Universität Duisburg-Essen ∗ , Technische Universität Darmstadt † , Universität Karlsruhe (TH) ‡ 01. April 2008 GAMM 2008 - Bremen Symmetry of Biot-stresses and Biot-strains 01. April 2008 1 / 19

Introduction Overview Section 1 Introduction 2 From Biot to Cosserat 3 Symmetry of Cauchy-stresses and Biot-strains 4 Conclusions Symmetry of Biot-stresses and Biot-strains 01. April 2008 2 / 19

Introduction Notation Deformation of a Body • Reference configuration: Ω ⊂ R 3 • Deformation ϕ : Ω → Ω def ⊂ R 3 • Deformation gradient: F := ∇ ϕ ∈ GL + ( 3 ) Right Polar Decomposition: F = RU = polar ( F ) U (exact rotation of R 3 ) • Polar rotation: R , polar ( F ) ∈ SO ( 3 ) • Biot-stretch: U ∈ PSym ( 3 ) (positive definite symmetric) A Crucial Identity √ • R T F = U = F T F ∈ PSym ( 3 ) Symmetry of Biot-stresses and Biot-strains 01. April 2008 3 / 19

Introduction Hyperelasticity, Objectivity and Material Isotropy Hyperelasticity - Minimizing total stored energy � Find minimizers ϕ min of: I ( ϕ ) = W ( ∇ ϕ ) dV Ω • Choice of W : M ( 3 , R ) → R is a constitutive assumption • Can apply various boundary conditions ... • Consider only zero body forces Isotropy and Objectivity in terms of F := ∇ ϕ • W ( F ) is objective if ∀ Q ∈ SO ( 3 ) : W ( QF ) = W ( F ) • W ( F ) is isotropic if ∀ Q ∈ SO ( 3 ) : W ( FQ ) = W ( F ) Objectivity of W ( F ) ⇒ ∃ W ♯ : PSym ( 3 ) → R s. th. W ( F ) = W ♯ ( U ( F )) Symmetry of Biot-stresses and Biot-strains 01. April 2008 4 / 19

From Biot to Cosserat Overview Section 1 Introduction 2 From Biot to Cosserat 3 Symmetry of Cauchy-stresses and Biot-strains 4 Conclusions Symmetry of Biot-stresses and Biot-strains 01. April 2008 5 / 19

From Biot to Cosserat The isotropic Biot-model - Standard version Basic constitutive assumptions √ • Objectivity: W ( F ) = W ♯ ( U ) , where U := R T F = F T F ∈ PSym • Isotropy: ∀ Q ∈ SO ( 3 ) : W ( FQ ) = W ( F ) ⇐ ⇒ W ♯ ( Q T UQ ) = W ♯ ( U ) Example: Most general isotropic quadratic energy in U √ W ♯ ( U ) = µ � U − 1 � 2 + λ 2 tr [ U − 1 ] 2 , U := F T F • Linearization equivalent to classical isotropic linear elasticity • µ, λ the standard Lamé moduli √ Biot approach is intrinsically based on a formulation in U = F T F • Have to take derivatives of U , i.e., of a matrix square root T F seems to be convenient • Numerical relaxation U := R Symmetry of Biot-stresses and Biot-strains 01. April 2008 6 / 19

From Biot to Cosserat Euler-Lagrange equations for the Biot-model Free variation w.r.t. to ϕ � � � � d � D F [ W ♯ ( U ( F ))] , ∇ v � dV = � D U W ♯ ( U ) , D F U ( F ) . ∇ v � dV 0 = � I ( ϕ + t v ) = � D F W ( ∇ ϕ ) , ∇ v � dV = dt � t = 0 Ω Ω Ω � � � D U W ♯ ( U ) , D F [ R ( F ) T F ] . ∇ v � dV = . . . = � R ( F ) D U W ♯ ( U ) , ∇ v � = Ω Ω � � Div [ R ( F ) D U W ♯ ( U )] , v � dV , (Ω , R 3 ) . = ∀ v ∈ C ∞ 0 Ω Strong form of the equilibrium equation 0 = Div [ R ( F ) D U W ♯ ( U )] , R ( F ) = polar ( F ) A weaker form of the equilibrium equation 0 = Div [ R ( F ) D U W ♯ ( R ( F ) T F )] , R ( F ) T F ∈ Sym Symmetry of Biot-stresses and Biot-strains 01. April 2008 7 / 19

From Biot to Cosserat Euler-Lagrange equations for the Biot-model Free variation w.r.t. to ϕ � � � � d � D F [ W ♯ ( U ( F ))] , ∇ v � dV = � D U W ♯ ( U ) , D F U ( F ) . ∇ v � dV 0 = � I ( ϕ + t v ) = � D F W ( ∇ ϕ ) , ∇ v � dV = dt � t = 0 Ω Ω Ω � � � D U W ♯ ( U ) , D F [ R ( F ) T F ] . ∇ v � dV = . . . = � R ( F ) D U W ♯ ( U ) , ∇ v � = Ω Ω � � Div [ R ( F ) D U W ♯ ( U )] , v � dV , (Ω , R 3 ) . = ∀ v ∈ C ∞ 0 Ω Strong form of the equilibrium equation 0 = Div [ R ( F ) D U W ♯ ( U )] , R ( F ) = polar ( F ) A weaker form of the equilibrium equation 0 = Div [ R ( F ) D U W ♯ ( R ( F ) T F )] , R ( F ) T F ∈ Sym T F ∈ Sym T F ∈ PSym • Weaker means: R �⇒ U = R Symmetry of Biot-stresses and Biot-strains 01. April 2008 7 / 19

From Biot to Cosserat Numerical Relaxation Idea - Generalize a classical model, but preserve its solutions √ • U := R T F = polar ( F ) T F = F T F constrains rotation Relaxation of the constraint • Independent rotations R : Ω → SO ( 3 ) (no physics involved!) T F • Yields relaxed Biot-stretch U := R (usually not symmetric) • Fixing R = polar ( F ) recovers the non-relaxed Biot-case Consequences • Can define relaxed energies W ♯ ( U ) = W ( F , R ) • Relaxation gets rid of U but introduces R ∈ SO ( 3 ) • One obtains a 2-field model! Symmetry of Biot-stresses and Biot-strains 01. April 2008 8 / 19

From Biot to Cosserat The relaxed isotropic Biot-model is a Cosserat-model Hyperelastic Cosserat-model (Neff, 2006) � I ( ϕ, R ) := W mp ( F ( x ) , R ( x )) + W curv ( R ( x ) , D x R ( x )) dV �→ min. w.r.t. ( ϕ, R ) Ω Definition of the energetic components: Isotropic case W mp ( U ) := W shear ( U ) + W vol ( det [ U ]) W shear ( U ) := µ � sym ( U − 1 ) � 2 + µ c � skew ( U − 1 ) � 2 W vol ( det [ U ]) := λ − 1 − 1 ) 2 � � ( det [ U ] − 1 ) 2 + ( det [ U ] 4 T D x R 2 � W curv ( R , D x R ) := µ L 2 c � R Notation T F , � � U := R F := ∇ ϕ, D x R := ∇ ( R . e 1 ) , ∇ ( R . e 2 ) , ∇ ( R . e 3 ) Symmetry of Biot-stresses and Biot-strains 01. April 2008 9 / 19

From Biot to Cosserat The relaxed isotropic Biot-model is a Cosserat-model Hyperelastic Cosserat-model without curvature energy ˆ = L c = 0 � ❳❳ ✘ I ( ϕ, R ) := W mp ( F ( x ) , R ( x )) + ✘✘ W curv ( R ( x ) , D x R ( x )) dV �→ min. w.r.t. ( ϕ, R ) ❳ Ω Definition of the energetic components: Isotropic case W mp ( U ) := W shear ( U ) + W vol ( det [ U ]) W shear ( U ) := µ � sym ( U − 1 ) � 2 + µ c � skew ( U − 1 ) � 2 − 1 − 1 ) 2 � W vol ( det [ U ]) := λ � ( det [ U ] − 1 ) 2 + ( det [ U ] 4 T D x R 2 � W curv ( R , D x R ) := µ L 2 c � R ← W curv is absent if L c = 0 Notation T F , � � U := R F := ∇ ϕ, D x R := ∇ ( R . e 1 ) , ∇ ( R . e 2 ) , ∇ ( R . e 3 ) Symmetry of Biot-stresses and Biot-strains 01. April 2008 9 / 19

From Biot to Cosserat The relaxed isotropic Biot-model is a Cosserat-model Hyperelastic Cosserat-model without curvature energy ˆ = L c = 0 � I ( ϕ, R ) := W shear ( F , R ) + W vol ( F , R ) dV �→ min. w.r.t. ( ϕ, R ) Ω Definition of the energetic components: Isotropic case W mp ( U ) := W shear ( U ) + W vol ( det [ U ]) W shear ( U ) := µ � sym ( U − 1 ) � 2 + µ c � skew ( U − 1 ) � 2 W vol ( det [ U ]) := λ − 1 − 1 ) 2 � � ( det [ U ] − 1 ) 2 + ( det [ U ] ← constant w.r.t. R 4 T D x R 2 � W curv ( R , D x R ) := µ L 2 c � R ← W curv is absent if L c = 0 Notation T F , � � U := R F := ∇ ϕ, D x R := ∇ ( R . e 1 ) , ∇ ( R . e 2 ) , ∇ ( R . e 3 ) Symmetry of Biot-stresses and Biot-strains 01. April 2008 9 / 19

From Biot to Cosserat The relaxed isotropic Biot-model is a Cosserat-model ❍❍ ✟ Hyperelastic Cosserat-model with L c = 0 and ✟✟ W vol ❍ � T F − 1 ) � 2 + µ c � skew ( R T F − 1 ) � 2 dV �→ min. w.r.t. ( ϕ, R ) I ( ϕ, R ) := µ � sym ( R Ω Definition of the energetic components: Isotropic case W mp ( U ) := W shear ( U ) + W vol ( det [ U ]) W shear ( U ) := µ � sym ( U − 1 ) � 2 + µ c � skew ( U − 1 ) � 2 W vol ( det [ U ]) := λ − 1 − 1 ) 2 � � ( det [ U ] − 1 ) 2 + ( det [ U ] ← constant w.r.t. R 4 T D x R 2 � W curv ( R , D x R ) := µ L 2 c � R ← W curv is absent if L c = 0 Notation T F , � � U := R F := ∇ ϕ, D x R := ∇ ( R . e 1 ) , ∇ ( R . e 2 ) , ∇ ( R . e 3 ) Symmetry of Biot-stresses and Biot-strains 01. April 2008 9 / 19

From Biot to Cosserat The linear setting - σ ∈ Sym ⇐ ⇒ skew ( ∇ u − A ) = 0 Linearizations of the fields Linearized “constraint” • Finite: R = polar ( F ) • F = 1 + ∇ u , ∇ u ≪ 1 • Linear: A = skew ( ∇ u ) • R ≈ 1 + A , where A ∈ so ( 3 ) • U − 1 ≈ ∇ u − A • polar ( F ) ≈ 1 + skew ( ∇ u ) Definition of W ♯ - Cosserat without curvature W ♯ ( U ) := µ � sym ( U − 1 ) � 2 + µ c � skew ( U − 1 ) � 2 � � + λ 1 ( det [ U ] − 1 ) 2 + ( det [ U ] − 1 ) 2 4 Symmetry of Biot-stresses and Biot-strains 01. April 2008 10 / 19

From Biot to Cosserat The linear setting - σ ∈ Sym ⇐ ⇒ skew ( ∇ u − A ) = 0 Linearizations of the fields Linearized “constraint” • Finite: R = polar ( F ) • F = 1 + ∇ u , ∇ u ≪ 1 • Linear: A = skew ( ∇ u ) • R ≈ 1 + A , where A ∈ so ( 3 ) • U − 1 ≈ ∇ u − A • polar ( F ) ≈ 1 + skew ( ∇ u ) W ♯ quad - Quadratic approximation of W ♯ quad ( ∇ u − A ) := µ � sym ( ∇ u − A ) � 2 + µ c � skew ( ∇ u − A ) � 2 + λ � 2 W ♯ � 2 tr ∇ u − A Linear case µ c > 0 : σ ∈ Sym ⇐ ⇒ skew ( ∇ u − A ) = 0 σ = D ∇ u W ♯ quad = 2 µ sym ( ∇ u ) + 2 µ c skew ( ∇ u − A ) + λ tr [ ∇ u ] 1 Symmetry of Biot-stresses and Biot-strains 01. April 2008 10 / 19